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Yes it should. And the diagram only works when f is increasing ..... so it's not a fully rigorous proof.
Thanks Bob. That was very interesting (and I actually understood what you were saying).
Shouldn't it be less than or equal to in the inequalities?
hi amberzak and Stefy,
So F is made by just summing the areas of all the infinitely thin rectangles with height f(x) and width dx.
Now, on the f(x) graph I have shaded an area in yellowy orange to indicate the area under f up to a vertical line x.
Then I have added a thin strip in red , width h (delta x) , for the next little bit of area to be added.
I've used h on the diagram because getting a delta x is a bit tricky on this diagram.
So how big is this extra area? Well it is smaller than a rectangle width h and height f(x+h).
And it is bigger than a rectangle width h and height f(x)
so let delta x tend to zero
so the differential gets sandwiched between two values that approach each other and in the limit
So differentiation is the reverse of integration.
I'm going to ask why Bob. Because I always want to know why.
Oh,yes.I forgot that! Ok,thanks for the clarification,K5.
You are not integrating between the two crossing points on the x axis. Integration theory works out areas between two ordinates. Drive those little areas from your mind completely.
If you ask, I'll go through the 'why' for you.
Anon, I was wondering that as well, actually. I might ask my teacher than tomorrow.
Thanks guys. That's really helpful.
I saw it. I still do not get why the two small parts next to the rectangle don't mess it up.
Ahead of you for the first time. see post 20
No,he didn't integrate the term 2x correctly.
I see it now. I've spent all day driving and gardening (big tree to cut down) so my brain is not at its best. Here is a correction to post #12.
= 18 and 2/3
less 8 = 10 and 2/3
My apologies for the error before. I need some sleep.